In this talk, I will discuss a series of works on Gromov's p-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index p. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the p-widths and how surfaces provide a convenient setting to investigate this. 

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has at most $C$-quadratic decay at infinity for some $C > \frac{2}{3}$, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and $\mathbb{S}^2\times \mathbb{S}^1$ summands. Consequently, $M$ carries a complete Riemannian metric of uniformly positive scalar curvature. The decay constant $\frac{2}{3}$ is sharp, as demonstrated by metrics on $\mathbb{R}^2 \times \mathbb{S}^1$. This improves a result of Balacheff, Gil Moreno de Mora Sard{\`a}, and Sabourau, and partially answers a conjecture of Gromov. The main tool is a new exhaustion result using $\mu$-bubbles.

In this talk, we discuss recent developments in ancient solutions to the mean curvature flow in
higher dimensions. Consider an ancient flow asymptotic to a cylinder with the number of R factors equal
to k, we show that the asymptotic behavior of the flow is characterized by a k x k matrix Q whose
eigenvalues can only be 0 and 1. We further discuss the cases where Q is fully degenerate or fully
nondegenerate under the noncollapsing assumption. In the fully degenerate case, we obtain a complete
classification. In the fully nondegenerate case, we establish a rigidity result showing that the solutions are
determined by only k-1 parameters. This is based on joint work with Beomjun Choi and Wenkui Du.

A central question in geometric flows is to understand the formation of singularities. In this talk, I will focus on mean curvature flow, the negative gradient flow of area functional, and explain how the local dynamics influence the shape of the flow and its singular set near a cylindrical singularity, as well as how the topology of the flow changes after passing through such a singularity with generic dynamics. This talk is based on the joint works with Ao Sun and Jinxin Xue.

Mean curvature flow (MCF) is the deformation of surfaces with velocity equal to the mean curvature vector. MCF originated in materials science and is widely used as a tool for geometric and topological problems. Major open questions about MCF include how large of singular sets can form, whether the area of the flow is continuous through singular times, and how the various weak solutions may differ. We address these questions under an assumption on the size of the set of singularities with “slow” mean curvature growth. With this assumption, an n-dimensional flow has H^n-measure zero singular sets at every time, has mass that is continuous through singular times, and under an additional mild condition, the level set flow fattens at the discrepancy time of the inner/outer flow. The key technical development is a generalized Brakke equality, which characterizes the deviation from equality in Brakke’s inequality. This is achieved by developing a worldline analysis of Brakke flow, which allows us to relate the regular parts of the flow at different times and estimate the transport of mass into and out of the singular set.